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Groups whose bipartite divisor graph for character degrees has four or fewer vertices

Faculty of Basic Science, University of Qom, Qom, Iran

https://doi.org/10.1142/S1793557120500394Cited by:1 (Source: Crossref)

Abstract

Let $G$ be a finite group and ${cd}^{*} (G)$ be the set of nonlinear irreducible character degrees of $G$ . Suppose that $ρ (G)$ is the set of primes dividing some elements of ${cd}^{*} (G)$ . The bipartite divisor graph for ${cd}^{*} (G)$ , $B (G)$ , is a graph whose vertices are the disjoint union of $ρ (G)$ and ${cd}^{*} (G)$ , and a vertex $p \in ρ (G)$ is connected to a vertex $a \in {cd}^{*} (G)$ if and only if $p | a$ . In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.

Communicated by A. F. Vasil’ev

Keywords:

AMSC: 20C15, 20D60

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